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Tomographic inversion of satellite photometry.
Part 2
Stanley C. Solomon, Paul B. Hays, and Vincent J. Abreu
A method for combining nadir observations of emissionfeatures in the upper atmosphere with the result of a
tomographic inversion of limb brightness measurements is presented. Simulated and actual results are
provided, and error sensitivity is investigated.
1.
Introduction
In an earlier paper' we described an algorithm for
recovering the 2-D volume emission rate of optical
emissions in the upper atmosphere from limb brightness measurements made by an orbiting spacecraft.
This work was based on an integral equation solved by
Cormack2 which determines the relationship between
a function and its line integrals. We simplified his
result to obtain
1
Gn(r) =--|
f
_
_ _
__
__n_
/9GM H vl
dp,
(1)
where Gn(r) is the nth Fourier coefficient with respect
to of the function g(r,O),Fn(p) is the nth Fourier
coefficient with respect to of the line integral function f(p,O),and Tn is the nth Chebyshev polynomial.
This formula may be used to evaluate the volume
emission rate function g of an optically thin emission
from measurements of its column brightness f taken at
various tangent heights and tangent angles p, 0 (Fig.
1). Since p > r in this expression, measurements taken
when the line of sight tangent point lies below the
region of interest are not suitable for this type of calculation.
In practice there is an upper limit to the number of
Fourier coefficients which may be retained in performing the inversion without encountering spurious oscil-
The authors are with University of Michigan, Space Physics Re-
search Laboratory, Department of Atmospheric & Oceanic Science,
Ann Arbor, Michigan 48109-2143.
Received 5 June 1985.
0003-6935/85/234134-07$02.00/0.
©1985 Optical Society of America.
4134
APPLIED OPTICS / Vol. 24, No. 23 / 1 December 1985
lations. The limit depends on the instrumental characteristics, photometer counting statistics, temporal
variations of the emission, and distance below the satellite orbit. This places a constraint on the angular
resolution which may be obtained. In the case of the
Visible Airglow Experiment
3
(VAE) on the Atmo-
sphere Explorer (AE) satellites, about 20 to 30 Fourier
coefficients may be included before oscillatory behavior begins. This inversion is an improvement over
those previously applied to these data4 5 but still does
not attain the resolution we desire for imaging auroral
emissions and other thermospheric phenomena possessing high spatial variability.
When the AE satellites were in the spinning mode,
cartwheeling along the satellite track so that the photometer swept out a full circle in the orbital plane, limb
scans were obtained which are amenable to interpretation by the Cormack inversion. Information about the
emission function is also acquired while the photometer points downward at the earth, but these measurements are contaminated by light scattered from the
lower atmosphere, clouds, and solid earth below. In
the daytime, these downward looking observations are
useless, but at night the contribution from scattered
light may be calculated and subtracted from the
data. 6
7
When the photometer line of sight intersects lower
atmosphere and ground, the measurement no longer
represents a complete line integral of the upper atmosphere emission function, and straightforward inversion techniques no longer apply. Rather than attempting to interpret all these observations, we have
elected to incorporate into the algorithm only those
made looking directly down at the earth within a small
angle of the vertical. These nadir observations may be
combined with the result of the Cormack inversion to
increase the horizontal resolution of the recovery. A
method for accomplishing this is described below, simulated results are presented, and selected VAE orbits
are inverted. In addition, we present an analysis of
errors engendered by the data reduction.
II.
Theory
Nadir photometry yields a measured vertical column brightness function b(O) where 0 is the angle
along the satellite track. This function is the sum of
light emitted by the upper atmosphere ba(O)and scattered by the lower atmosphere and ground bg(0). During nighttime conditions, and assuming that there are
no other significant sources of illumination, bg(O)depends on ba(O),the planetary albedo a(0), the height of
the emitting layer Za,and the scattering phase function. If we may assume that the scattering is Lambertian, that the emitting layer has insignificant vertical
extent and is a function only of 0, that a plane parallel
atmosphere adequately approximates the actual case,
and that the albedo is constant, then6
ba(O)= b(O) -
b
2J
(
b,(O')w-'(0
- ')dO',
2)
(2)
Fig. 1.
Inversion geometry. Each line integral f is the sum of all
g(r,O) along a path specified by distance p and angle p.
where the 1-D inverse weighting function w- 1 is de-
fined
w-'(p) =
fI
exp(-kza) cos(kp) dk.
1 + 2a exp(-kz.)
7rJ0
the Cormack inversion. For n > nmax, G(r) can be
computed from Ban
(3)
The approximations involved are reasonable excepting that in some cases there may be variations outside
the orbital plane, violating the assumption that ba is a
function only of 0, and that the albedo may vary. The
latter is potentially serious, but in certain cases of
interest such as the winter polar regions, clouds, snow,
and ice all have similar reflectivities so we may hope
that albedo variations will be small.
Assuming that the albedo is constant, it may be
estimated by the following procedure. The volume
emission rate in photons cm-3 sec-1 recovered by the
Cormack inversion is integrated over the r,0 plane; this
quantity I, must be equal to the nadir column brightness in 10-6 R integrated over 0:
ba(O) =fJ
Ban
b,,,(O)dO-
J
bg(O)dO
2.r
f
K(r)
ba(O)dO,
-
bm()dO -I
2I,
(9)
(10)
where
r27r
j
(8)
R(r)dr,
Gn(r) = K(r)Ba,n,
(4)
since the integral of the 1-D direct weighting function6
equals 2a. Therefore,
J
Therefore,
2r
b.(O)dO- 2a
on
(7)
Gn(r) = R(r)On-
r2~~~~~
=
=
g(r,O)dO,
B(r) a.,O
(11)
Furthermore, if the shape of the r dependence varies
slowly with 0, information as to the nature of this
variation is contained by the low-order terms of Gn(r).
If the r dependence of coefficients greater than nmax
have the same shape, we have a situation where the
Gn(r) calculated from limb scans provide all the information about g(r,0) except the high frequency variation with respect to 0, which is contained in the Ba,n It
is then possible to splice the normalized Ba,nonto the
Gn(r) at each r to obtain a set of coefficients from which
g(r,0) may be evaluated using an inverse Fourier transform.
~~~~~~~(5)
Rather than abruptly
Equation (2) is a convolution integral and so may be
expressed in frequency space:
switching from Gn(r) to Ba,nat
nmax,it is desirable to effect a smooth transition from
one set of coefficients to the other. This may be accomplished using error functions:
Ba n =B,n -BmnWn1
Gn'(r) = ElGn(r) + E2K(r)Ba,n,
2I4
(6)
where upper case letters represent the Fourier transform with respect to 0 of the corresponding function,
and n is the Fourier coefficient index. Suppose that
any point in the r,0 plane, g(r,0) = R(r)0(0); that is, the
r dependence has the same shape at all 0. For n • nmax,
Gn(r) can be evaluated using Eq. (1) where nmaxis the
maximum number of terms that can be retained using
(12)
where
E1
+ erf(n,
-
n)/2]
2
1+ erft(n- n)/2]
=
E2
-
2
1 December 1985 / Vol. 24, No. 23 / APPLIED OPTICS
(13)
(14)
4135
_111 111 11
value
T
1
MIM111
Illllll
plotted in Fig. 2 for the case n, = 15, the value used in
the inversions presented below.
I-
Equation (10) will be strictly valid only so long as the
E
I
condition concerning the shape of the r dependence
holds. If this condition is approximately satisfied, Eq.
(10) may still be an acceptable estimate, particularly in
the region of maximum brightness. When the volume
emission rate function recovered using Eq. (1) exhibits
significantly varying altitude profiles, caution in using
Eq. (12) to find Gn'(r) is indicated. Also, the spectral
resolution of Gn'(r) is still limited by the Ba(0);in the
case of VAE data, this results in a maximum resolution
in 0 of about one degree. Thus this technique is best
suited to the analysis of isolated stratified emission
features of significant horizontal extent.
2
. 1 1 1... .. 1
I. . . .111. I.1 1. 1. 11
0
20
10
Foui-
Fig. 2.
coefficient
Transition weighting functions.
erf(x) = 27r 1 /2
J:
2
exp(-t
)dt,
(15)
of
111. Simulation and Error Analysis
and n, is the Fourier term at the center of the transition. The transition weighting functions E1 and E2 are
300
I
I
.
I
.
I
.
,
I
I
1
I
.
I
I
I
The random and systematic differences between an
actual emission function and that which is garnered
200_
160
photons
/ccsec
(a)
Altit. iude
(kM.
Emsisstn
__(d)
photons
-
I.
120
200
/ /e100
a
100
40
.i
I
-
0I
I
I
10
20
30
Igle
.
I
.
I
liii
40
Along Track
.
I
.
I
ii
50
60
70
0
0
90
(degrees)
.
I
.
single Along Track
I
.
I
.
I
(degrees)
160
photons
(b)
Altitude
(kM)
10/sec
-
120
Altitude
(km)
200
80
I ;.
3.
40
100 -
0
O
1
10
220
30
30
I
50
40
Rng1e Along Track
I
300
,
I
,
I
.
I
90
.
0
9
E.sission
(degrees)
.
.
I . 70
60
I
.
II
.
.
I
I
_w
160
I
.
,
120
I 111111
§
Altitude
(km)
-
X
X
i- .,
.,i,.S,7
200-
x
x
_
100
.1.11.,1. .
(photons'cc'sec)
1
.
.
1
Illlli
1
......
1
I
.
I
1 111111
iAS--0.
X
X
+
l
. -
I
photons
/00/sec
(c)
Altitude
(km)
200
Rate
.....
+
x+
. ..
x
.40
10
x
x
+>
+x
0I
I0
10
0
20
30
40
Age
Along Track
50
60
70
00
90
(degrees)
100
II
r
T
I
10
Emission Rate
Fig. 3. (a) Simulated emission function. (b) Inversion using only
limb scans. (c) Inversion using limb scans and nadir observations.
(d) Emission rate at 150 km. (e) Emission rate altitude profile
through peak of left feature. (f) Emission rate altitude profile
through peak of right feature. Solid line-simulation; crosses
-limb scan inversion; X-including nadir.
4136
APPLIED OPTICS / Vol. 24, No. 23 / 1 December 1985
r
Irl
T
Ir
I 1r
100
(photonscosec)
I
1000
from an inversion algorithm are best deduced through
computational simulation. This is accomplished by
specifying a 2-D function having characteristics similar to those we expect of an upper atmosphere emission
and calculating its line integrals numerically at intervals typical of those employed by the VAE channel 1
photometer.1 3 The instrumental field of view was
simulated by weighted averaging of seven line integrals
arrayed in 0.30 intervals around the central line of
sight. Gaussian noise equal to the square root of the
number of counts was also added using a characteristic
sensitivity equal to 0.1 counts/R. Nadir observations
Emissi on
Rate
photons
'cc/sec
100
Rgle
were handled similarly, integrating over a window of
300
100halfwidth, with scattered light calculated using the
direct weighting function6 and added to the simulated
brightness. Several emission functions were utilized
to examine the effect of sharp variations in the horizontal and vertical directions, and situations violating
1,
Along Track
i .11
1
_+
X
+
Rltit
(ke
x
x
xX
+ +
+
200+
+
10
I
Emission
Fig. 4.
H
(b)_
+
some of the conditions mentioned in Sec. II were em-
ployed to evaluate the algorithm's sensitivity to such
deviations.
Figure 3(a) displays two imaginary emission functions, both generated by multiplying a Chapman function in altitude by a Gaussian in angle along the track.
(The Chapman function has the form expf1 + (rp r)IH - exp[(rp - r)/I]}, where rp is the peak altitude
and H is the scale height.) The feature on the left is
based on a scale height of 30 km with the peak at 150
km, while the one on the right has a 15-km scale height
with the peak at 120 km, so the total emission does not
satisfy the requirement for applying nadir observations that the variables be separable in the low frequencies. An albedo of 0.8 was used in computing the
nadir brightnesses. In Fig. 3(b) the limb scan inversion is displayed, and the inversion including nadir
observations is shown in Fig. 3(c). Despite the fact
that the function is not in compliance with our criteria,
some improvements are apparent. The addition of
nadir data has eliminated most of the horizontal
smearing of the recovery, as can be seen in Fig. 3(d),
which is a plot of the simulated and recovered emission
rates at 150km. The altitude profiles are displayed in
Figs. 3(e) and (f) for the left and right features, respectively. The increase in angular resolution afforded by
the nadir observations also improves the recovered
altitude dependence of the left feature, but the right
feature's profile is distorted by the technique: Although the peak value is closer to the original, at higher
altitudes the recovered emission rate is too large. This
is because the other feature is dominant at these
heights, and the inversion imposes its characteristics
on the fainter emission. Neither technique can resolve the sharp altitude dependence at the peak as they
are both limited by the VAE field of view and integration period.
To analyze these effects separately, the following
simulations are presented. A function which is narrow
horizontally but broad vertically (50-km scale height)
was simulated and inverted; again an albedo of 0.8 was
employed for the nadir simulations. Figure 4(a) demonstrates the great improvement gained by applying
,,II ,,1
\
+
ude
(degrees)
1
Rate
.
X
X
t00
1000
(photons/cosec)
Simulated inversion of a narrow feature:
(a) horizontal
variation at 150 km, (b) altitude profile at peak. Solid line-simulation; crosses-limb scan inversion; X-including nadir.
_11
_
Altit,
(km
ide
Il
j
1111
_
II II
|X|T|W
-i|
_
200
I
*
+~<
10
Emission Rate
100
1000
(photons/co/sec)
Fig. 5. Altitude profile of a wide, vertically structured, simulated
emission: solid line-simulation;
crosses-limb
scan inversion;
X-including nadir.
nadir observations in cases of this type as the feature is
much too narrow to be resolved by thirty Fourier coefficients. The altitude profile is also improved-the
limb scans ascertain the correct shape, but at the peak
of the emission the recovery is much too low unless
nadir data are applied as seen in Fig. 4(b). When the
emission is wide horizontally, limb scans alone are
sufficient for reconstruction. Figure 5 shows such a
case where a two-layered altitude dependence is simulated. The higher broader (30-km scale height) layer
is accurately reproduced, but the lower sharper (15-km
scale height) layer is smeared by instrumental broadening.
The simulations presented in Fig. 6 are all based on a
parabolic altitude profile with the peak at 200 km and a
1 December 1985 / Vol. 24, No. 23 / APPLIED OPTICS
4137
_
300
_Illl1
1 1 1 11H
Gm)
Altitude
(km)
Altil ude_
(k
200
I111
100~ _
ii
I10
10
Emission Rate
300
I
100
_ 1
_I|F-1
_
1
Fig. 7.
Ml
_
200
100
10
Emts
Rate
ion
_
1000
(photonsocc/sec)
zoo_ I 111111111
Us
+
100
Rate
.
}
t
l-Jl
l:1111
1111
S
~~~.
>
+
phot
'cc'
io
00
1
llll
1
ll111
+lii
-
200-
100
l
I
I
l
l
l 111111 100
Rate
l
l
I 111111
1000
10
Emission
Fig. 6.
l
l 111111
(photonscc/sec)
Simulated inversion of a parabolic altitude profile: (a) 0.1-
counts/R sensitivity; (b) 0.001-counts/R sensitivity; (c) emission
rate at 200 km including variable albedo case; (d) altitude profile for
variable albedo case. Solid line-simulation;
crosses-limb scan
inversion; X-including nadir; dotted line-including nadir, variable albedo.
square horizontal dependence. At the center of the
feature, either method does an excellent job of recovering the altitude dependence. Figure 6(a) shows an
inversion using nadir data. In Fig. 6(b) the effect of
noise is investigated by reducing the simulation instrument sensitivity by a factor of 100. The altitude pro4138
Simulated inversion as in Fig. 4 but with wrong albedo:
file recovered now oscillates about the actual one, deviating by as much as 15% from the original. This
simulation is equivalent to the case where the VAE
photometer observes a faint emission of 1-2 photons
cm 3 sec 1, such as the 5200-Aauroral line. For emissions of this type, nadir data should not be used as it
includes stray light from the much brighter lines at
lower altitude. The effect of noise can be mitigated by
smoothing the limb scans before subjecting them to
the inversion procedure. Next we investigate the effect of a varying albedo. Figure 6(c) shows the emission rate at 200 km recovered when the albedo switches
from 0.8 to 0.2and back every 30;this might be encountered, for example, if there were banded cloud structures over the open ocean. The albedo estimation
technique yields a constant value of 0.53, and hence the
recovery fluctuates as much as 25% above and 40%
below the simulation value. Also displayed in Fig. 6(c)
(d)
Altit.Ide
(km)
50
(degrees)
solid line-simulation; dotted line-inversion including nadir.
(b)
,
Alti1 Lude__
_
(kc
Emission
40
Along Track
ngle
(photonsoccsec)
1il 1I11
1
30
1000
APPLIED OPTICS / Vol. 24, No. 23 / 1 December 1985
is the result when the albedo is constant at 0.8 and
when limb scans alone are used. The altitude profile
from the highest fluctuation is plotted in Fig. 6(d) for
comparison with the original.
A case where the simulation albedo is constant at 0.8
but the albedo used in the inversion is only 0.4follows.
This situation might arise if the albedo estimation
method cannot be applied because the ground is illuminated by sunlight at the edge of the region of interest, or if temporal variations in the emission cause an
unrealistic estimation of the albedo which must be
rejected. We may then guess the albedo and wish to
know the consequences of a bad guess. Figure 7 demonstrates that the consequences are not severe. Here
we have used the same simulation as in Fig. 4, and the
recovery is similar but slightly broader in the wings.
The peak value is reduced by 27% as compared with
19%with a correct albedo. The reason there is so little
effect from halving the albedo is that in the combined
inversion the integrated emission rate at each altitude
is given by the limb scans, and only the horizontal
shape is changed by the nadir data.
Finally, the first simulation (Fig. 3) is recalculated
with a linearly increasing time dependence. In such
conditions the nadir data will be in conflict with the
limb scans, since the photometer observes a feature on
the limb several minutes before it passes over it. Not
surprisingly, the albedo estimated is an impossible 1.2,
300
I
I
I(
I
I
I
I
Visible Airgiow Experiment
AE-C Orbit
3303 day 78070
I
250
XE_
Altitude
,_lIIIIlllilllilll
Volume Emission Rate
4Z78 Rmgstro-s
IIIIIIIIIIIIIIIII
300
-
3
h
(a
200
2
Altitude
(kn)
150
++
+
100
.1
II
+
+
100
X
X
X
XX \)
.1
t0
.
......
Ad
Emission Rate
TI
0
50
300
X
_
(~~~~~~~~~~~~~~b)
l
l llid
I
I
c
l l 1l
i
-b_
-A
"-X
+
+
+
I _I 1
c
100
65
86
4
39990
X
+
Alt itu
ki)
90
68
64
3
39820
66
79
1
39670
_ ++
Altitude
100
O
80
ngle Al ong Track
70
Latitud
61
Rag. InclI inat on
72
Local cIlar Time
0
39510
UIi-vers ia Time
(photonsoccosec)
o
!
50
.
t
200
._
...__... +
+X X
200
P
Wa
L50 -.
xX.
AI +
*.Xi.
+
L50
..,X
+
_
i
-
100
I
10
Emission
100
Rate
1000
10
1
photons/ccsec)
428
Fig. 8. Time-dependent simulation altitude profiles (see Fig. 3):
(a) left feature; (b) right feature. Solid line-simulation;
crosses
-limb scan inversion; X-including nadir.
so in performing the inversion it was adjusted to 0.73,
equal to the value found when time dependence was
not included. The altitude profiles through the centers of the two features at the time the satellite passes
Fig. 9.
A
Emission Rate
100
1000
photonsccsec)
(a) Inversion including nadir data of auroral N 2 + 1NG (0,1)
band. (b) Altitude profile through peak. Crosses-limb scan inversion; X-including nadir; dotted line-theoretical model.
variation causes some underestimation of the emission
rate, these inversions suffer essentially the same faults
as the time independent ones. It may be noted here
that we do have a method for ascertaining when temporal fluctuations may be a problem-the spinning
satellite actually obtains two sets of limb scans, one
looking forward and one looking back, and these can be
inverted independently and compared with each other
and with the nadir observations.
Figure 10(a) displays results using the 6300-A filter
which measures the intensity of the atomic oxygen
(3 P-1D) transition. Because the emission is highly
structured, we may suspect the possibility of spurious
artifacts. Hence the total electron energy flux measured by the Low-Energy Electron Experiment 9
(LEE) is plotted in Fig. 10(b). The close correspondence of the electron precipitation peaks with the
bright emissions lends confidence to the result. An
altitude profile is plotted in Fig. 10(c) with error limits
estimated from the simulation analysis described
above. As expected, the 6300-Aemission is at relatively high altitude in the aurora, since O(D) is rapidly
deactivated by collisions with N2 at lower altitudes.
IV.
V.
over them are plotted in Fig. 8. Although.the
time
Application to Photometric Data
Two nightside auroral transits of AE-C were selected to demonstrate the efficacy of the inversion algorithm. Results from orbit 23,303 using the 4278-A
filter which passes the (0,1) band of the N2+ 1NG
system are presented in Fig. 9(a). The altitude profile
of the brightest column is plotted in Fig. 9(b). It is
accompanied by the profile recovered using only limb
scans and a theoretical model profile8 for comparison.
Including the nadir observations sharpens the altitude
dependence at the peak at the function, bringing it
closer to the theoretical result. The remainder of the
deviation is attributable to the effects of the photometer field of view and integration period-at high satellite altitude (380 km in this case) these problems are
exacerbated when viewingan emission at substantially
lower altitude.
Discussion
The scheme we have developed for interpretation of
satellite photometry may be described as an ad hoc
solution to the problem of combining two different
types of measurements-those made looking sideways
at an emission and those made looking down at it. In
essence, the technique is to use the former to ascertain
the vertical dependence and the latter to quantify the
horizontal variation. The more nearly separable the
functional dependence on the two variables the higher
the validity of the result. To recover a function which
is strongly-nonseparable, or when the underlying albedo is highly variable, measurements taken at additional angles would have to be included, and an iterative
solution may be a better method. This would introduce ambiguities from statistical variations of the data
and possible temporal variations of the emission, ne1 December 1985 / Vol. 24, No. 23 / APPLIED OPTICS
4139
Visible
Airg1ow ExperIment
75039
FE-C Orbit 5374 d
Volum
Emission Rate
6300 I"i tm
00
... .. ... . ... .. .
..............................................
. ... ... . .. ... ... ... .. .. .. ................
300
............
250
. ... .. .. .. .
Altitude(kin)
200
no
. .. .. .. ...
.. .. . .. .. . .
.. ... .. .. ..
.. .. . .. .. .-
150
.. ..
-.1
I-
__ i! f
.. .. ...
NI f I Iffi if f I ff
. . ..
ll
80
66
63
1
23450
Ai,!g1e
Along Trak
70
LatItude61
M ag. Incl1nation
64
0
L ocal S olar TI me
Ihiversal Te
23300
f 1
:i!
1
fi
90
66G
80
3
23G00
0
I
cessitating smoothing operations and the application
of a priori information.
Such an approach may be
indicated for certain cases and may be the only ap-
10
proach for optically thick emissions, but for stratified
auroral emissions the analytical solution here described is likely preferable. It is certainly a consistent
and reliable way to obtain the altitude profile of isolated features, which is the primary scientific goal of our
present efrs
100
65
76
s
23750
2.0
200
1k)
Electro
Flux
This work was supported by NASA grant NAGW-
496 to the University of Michigan.
erg cu
1-/O.tr
1.6
100
630 A
Emission
Ratephotons.
/cc.se0
1
3200
I 1
23300
-1I
23400
I
1
23900
tLkiiver l Ti e
( e
23600
1123700
1 o0.o
23)0
d )
Alti1
(k1
200
too
~ ~
~
~
10
1
6300 A
E mission Rate
H II I
I l 11
100
(ph oton.c..
1000
ec
Fig. 10. (a) Inversion including nadir observations of auroral OI(QP
- 'D) line. (b) Horizontal variation at 200 km with LEE total
electron energy flux for comparison.
Solid line-emission
rate;
dotted line-electron flux. (c) Altitude profile at UT 23655 with
estimated error limits.
4140
APPLIED OPTICS / Vol. 24. No. 23 / 1 December 1985
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